A Sage interact to play with 2 by 2 games

2by2games

def Pure_Nash_Check(A,B,i,j):
    if i==0:
        if j==0:
            if A[0,0]<A[1,0] or B[0,0]<B[0,1]:
                return False
        if j==1:
            if A[0,1]<A[1,1] or B[0,1]<B[0,0]:
                return False
    if i==1: 
        if j==0:
            if A[1,0]<A[0,0] or B[1,0]<B[1,1]:
                return False
        if j==1:
            if A[1,1]<A[0,1] or B[1,1]<B[1,0]:
                return False
    return True
 
def Nash(A,B):
    r=[]
    #Identify all pure equilibria
    for i in range(2):
        for j in range(2):
            if Pure_Nash_Check(A,B,i,j):
                if i==0:
                    p1=[1,0]
                else:
                    p1=[0,1]
                if j==0:
                    p2=[1,0]
                else:
                    p2=[0,1]    
                r.append([p1,p2])
    #If not all equilibria found identify mixed equilibria (since in a 2 by 2 game only 1 mixed equilibria possible)
    if len(r)%2==0:
        var("p,q")
        p=solve(B[0,0]*x+B[1,0]*(1-x)==B[0,1]*x+B[1,1]*(1-x),x,solution_dict=True)[0][x]
        q=solve(A[0,0]*x+A[0,1]*(1-x)==A[1,0]*x+A[1,1]*(1-x),x,solution_dict=True)[0][x]
        p1=[p,1-p]
        p2=[q,1-q]
        r.append([p1,p2])
    return r        
 
@interact
def _(A=matrix(RDF,2,2,[4,1,1,2]),p=(slider(0,1,.05)),B=matrix(RDF,2,2,[2,1,1,4]),q=(slider(0,1,.05))):
    show("Utilities:")
    p1_util= lambda x,y: x*y*A[0,0]+x*(1-y)*A[0,1]+(1-x)*y*A[1,0]+(1-x)*(1-y)*A[1,1]
    p2_util= lambda x,y: x*y*B[0,0]+x*(1-y)*B[0,1]+(1-x)*y*B[1,0]+(1-x)*(1-y)*B[1,1]
    
    p_static=copy(p)
    q_static=copy(q)
    
    p=plot(p1_util(x,q),(x,0,1),legend_label='Player 1',thickness=3)
    if p1_util(0,q)==p2_util(p_static,0) and p1_util(1,q)==p2_util(p_static,1):
        p+=plot(p2_util(p_static,x),(x,0,1),legend_label='Player 2',color='red',thickness=2)
    else:
        p+=plot(p2_util(p_static,x),(x,0,1),legend_label='Player 2',color='red',thickness=3)
    
    p_max=max(flatten(list([list(a) for a in A])))
    p_max=max(p_max,max(flatten(list([list(b) for b in B]))))
    p.ymax(p_max)
    
    p_min=min(flatten(list([list(a) for a in A])))
    p_min=min(p_min,min(flatten(list([list(b) for b in B]))))
    p.ymin(p_min)
    
    p.axes_labels(['First strategy','Utility'])
    show(p)
 
    r=Nash(A,B)
    l=len(r)
    if l==1:
        show("Single equilibria:")
        show(r[0])
        return
    for e in r:
        if l!=1:          
            show("Equilibria number %s:"%(r.index(e)+1))
        show(e)